Question

12 multiple choice 1 point

1. which is true about triangles abc and def?

a. the triangles are congruent because $\frac{12 - 6}{4 - 2}=\frac{27 - 15}{9 - 5}$
b. the triangles are congruent because $\frac{12 - 6}{4 - 2}=\frac{15 - 27}{9 - 5}$
c. the triangles are similar because $\frac{6 - 12}{4 - 2}=\frac{27 - 15}{9 - 5}$
d. the triangles are similar because $\frac{12 - 6}{4 - 2}=\frac{27 - 15}{9 - 5}$
a
b
c
d

Explanation:

Step1: Recall similarity - congruence criteria

Similar triangles have proportional side - lengths. Congruent triangles have equal side - lengths. The given equations are about the ratios of side - length differences.

Step2: Analyze the ratios

For two non - vertical line segments in a coordinate plane, if the ratios of the changes in the y - coordinates to the changes in the x - coordinates are equal, the triangles formed are similar.
For the left - hand side of the ratio $\frac{12 - 6}{4 - 2}=\frac{6}{2}=3$.
For the right - hand side of the ratio $\frac{27 - 15}{9 - 5}=\frac{12}{4}=3$.
Since the ratios of the corresponding side - length differences (which represent the slopes of the lines containing the sides of the triangles) are equal, the triangles are similar.

Answer:

D. The triangles are similar because $\frac{12 - 6}{4 - 2}=\frac{27 - 15}{9 - 5}$